(*  Title:      HOL/Hahn_Banach/Subspace.thy

     Author:     Gertrud Bauer, TU Munich

 *)

 header {* Subspaces *}

 theory Subspace

 imports Vector_Space

 begin

 subsection {* Definition *}

 text {*

   A non-empty subset @{text U} of a vector space @{text V} is a

   \emph{subspace} of @{text V}, iff @{text U} is closed under addition

   and scalar multiplication.

 *}

 locale subspace =

   fixes U :: "'a\<Colon>{minus, plus, zero, uminus} set" and V

   assumes non_empty [iff, intro]: "U \<noteq> {}"

     and subset [iff]: "U \<subseteq> V"

     and add_closed [iff]: "x \<in> U \<Longrightarrow> y \<in> U \<Longrightarrow> x + y \<in> U"

     and mult_closed [iff]: "x \<in> U \<Longrightarrow> a \<cdot> x \<in> U"

 notation (symbols)

   subspace  (infix "\<unlhd>" 50)

 declare vectorspace.intro [intro?] subspace.intro [intro?]

 lemma subspace_subset [elim]: "U \<unlhd> V \<Longrightarrow> U \<subseteq> V"

   by (rule subspace.subset)

 lemma (in subspace) subsetD [iff]: "x \<in> U \<Longrightarrow> x \<in> V"

   using subset by blast

 lemma subspaceD [elim]: "U \<unlhd> V \<Longrightarrow> x \<in> U \<Longrightarrow> x \<in> V"

   by (rule subspace.subsetD)

 lemma rev_subspaceD [elim?]: "x \<in> U \<Longrightarrow> U \<unlhd> V \<Longrightarrow> x \<in> V"

   by (rule subspace.subsetD)

 lemma (in subspace) diff_closed [iff]:

   assumes "vectorspace V"

   assumes x: "x \<in> U" and y: "y \<in> U"

   shows "x - y \<in> U"

 proof -

   interpret vectorspace V by fact

   from x y show ?thesis by (simp add: diff_eq1 negate_eq1)

 qed

 text {*

   \medskip Similar as for linear spaces, the existence of the zero

   element in every subspace follows from the non-emptiness of the

   carrier set and by vector space laws.

 *}

 lemma (in subspace) zero [intro]:

   assumes "vectorspace V"

   shows "0 \<in> U"

 proof -

   interpret V: vectorspace V by fact

   have "U \<noteq> {}" by (rule non_empty)

   then obtain x where x: "x \<in> U" by blast

   then have "x \<in> V" .. then have "0 = x - x" by simp

   also from `vectorspace V` x x have "\<dots> \<in> U" by (rule diff_closed)

   finally show ?thesis .

 qed

 lemma (in subspace) neg_closed [iff]:

   assumes "vectorspace V"

   assumes x: "x \<in> U"

   shows "- x \<in> U"

 proof -

   interpret vectorspace V by fact

   from x show ?thesis by (simp add: negate_eq1)

 qed

 text {* \medskip Further derived laws: every subspace is a vector space. *}

 lemma (in subspace) vectorspace [iff]:

   assumes "vectorspace V"

   shows "vectorspace U"

 proof -

   interpret vectorspace V by fact

   show ?thesis

   proof

     show "U \<noteq> {}" ..

     fix x y z assume x: "x \<in> U" and y: "y \<in> U" and z: "z \<in> U"

     fix a b :: real

     from x y show "x + y \<in> U" by simp

     from x show "a \<cdot> x \<in> U" by simp

     from x y z show "(x + y) + z = x + (y + z)" by (simp add: add_ac)

     from x y show "x + y = y + x" by (simp add: add_ac)

     from x show "x - x = 0" by simp

     from x show "0 + x = x" by simp

     from x y show "a \<cdot> (x + y) = a \<cdot> x + a \<cdot> y" by (simp add: distrib)

     from x show "(a + b) \<cdot> x = a \<cdot> x + b \<cdot> x" by (simp add: distrib)

     from x show "(a * b) \<cdot> x = a \<cdot> b \<cdot> x" by (simp add: mult_assoc)

     from x show "1 \<cdot> x = x" by simp

     from x show "- x = - 1 \<cdot> x" by (simp add: negate_eq1)

     from x y show "x - y = x + - y" by (simp add: diff_eq1)

   qed

 qed

 text {* The subspace relation is reflexive. *}

 lemma (in vectorspace) subspace_refl [intro]: "V \<unlhd> V"

 proof

   show "V \<noteq> {}" ..

   show "V \<subseteq> V" ..

   fix x y assume x: "x \<in> V" and y: "y \<in> V"

   fix a :: real

   from x y show "x + y \<in> V" by simp

   from x show "a \<cdot> x \<in> V" by simp

 qed

 text {* The subspace relation is transitive. *}

 lemma (in vectorspace) subspace_trans [trans]:

   "U \<unlhd> V \<Longrightarrow> V \<unlhd> W \<Longrightarrow> U \<unlhd> W"

 proof

   assume uv: "U \<unlhd> V" and vw: "V \<unlhd> W"

   from uv show "U \<noteq> {}" by (rule subspace.non_empty)

   show "U \<subseteq> W"

   proof -

     from uv have "U \<subseteq> V" by (rule subspace.subset)

     also from vw have "V \<subseteq> W" by (rule subspace.subset)

     finally show ?thesis .

   qed

   fix x y assume x: "x \<in> U" and y: "y \<in> U"

   from uv and x y show "x + y \<in> U" by (rule subspace.add_closed)

   from uv and x show "\<And>a. a \<cdot> x \<in> U" by (rule subspace.mult_closed)

 qed

 subsection {* Linear closure *}

 text {*

   The \emph{linear closure} of a vector @{text x} is the set of all

   scalar multiples of @{text x}.

 *}

 definition

   lin :: "('a::{minus, plus, zero}) \<Rightarrow> 'a set" where

   "lin x = {a \<cdot> x | a. True}"

 lemma linI [intro]: "y = a \<cdot> x \<Longrightarrow> y \<in> lin x"

   unfolding lin_def by blast

 lemma linI' [iff]: "a \<cdot> x \<in> lin x"

   unfolding lin_def by blast

 lemma linE [elim]: "x \<in> lin v \<Longrightarrow> (\<And>a::real. x = a \<cdot> v \<Longrightarrow> C) \<Longrightarrow> C"

   unfolding lin_def by blast

 text {* Every vector is contained in its linear closure. *}

 lemma (in vectorspace) x_lin_x [iff]: "x \<in> V \<Longrightarrow> x \<in> lin x"

 proof -

   assume "x \<in> V"

   then have "x = 1 \<cdot> x" by simp

   also have "\<dots> \<in> lin x" ..

   finally show ?thesis .

 qed

 lemma (in vectorspace) "0_lin_x" [iff]: "x \<in> V \<Longrightarrow> 0 \<in> lin x"

 proof

   assume "x \<in> V"

   then show "0 = 0 \<cdot> x" by simp

 qed

 text {* Any linear closure is a subspace. *}

 lemma (in vectorspace) lin_subspace [intro]:

   "x \<in> V \<Longrightarrow> lin x \<unlhd> V"

 proof

   assume x: "x \<in> V"

   then show "lin x \<noteq> {}" by (auto simp add: x_lin_x)

   show "lin x \<subseteq> V"

   proof

     fix x' assume "x' \<in> lin x"

     then obtain a where "x' = a \<cdot> x" ..

     with x show "x' \<in> V" by simp

   qed

   fix x' x'' assume x': "x' \<in> lin x" and x'': "x'' \<in> lin x"

   show "x' + x'' \<in> lin x"

   proof -

     from x' obtain a' where "x' = a' \<cdot> x" ..

     moreover from x'' obtain a'' where "x'' = a'' \<cdot> x" ..

     ultimately have "x' + x'' = (a' + a'') \<cdot> x"

       using x by (simp add: distrib)

     also have "\<dots> \<in> lin x" ..

     finally show ?thesis .

   qed

   fix a :: real

   show "a \<cdot> x' \<in> lin x"

   proof -

     from x' obtain a' where "x' = a' \<cdot> x" ..

     with x have "a \<cdot> x' = (a * a') \<cdot> x" by (simp add: mult_assoc)

     also have "\<dots> \<in> lin x" ..

     finally show ?thesis .

   qed

 qed

 text {* Any linear closure is a vector space. *}

 lemma (in vectorspace) lin_vectorspace [intro]:

   assumes "x \<in> V"

   shows "vectorspace (lin x)"

 proof -

   from `x \<in> V` have "subspace (lin x) V"

     by (rule lin_subspace)

   from this and vectorspace_axioms show ?thesis

     by (rule subspace.vectorspace)

 qed

 subsection {* Sum of two vectorspaces *}

 text {*

   The \emph{sum} of two vectorspaces @{text U} and @{text V} is the

   set of all sums of elements from @{text U} and @{text V}.

 *}

 instantiation "fun" :: (type, type) plus

 begin

 definition

   sum_def: "plus_fun U V = {u + v | u v. u \<in> U \<and> v \<in> V}"  (* FIXME not fully general!? *)

 instance ..

 end

 lemma sumE [elim]:

     "x \<in> U + V \<Longrightarrow> (\<And>u v. x = u + v \<Longrightarrow> u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> C) \<Longrightarrow> C"

   unfolding sum_def by blast

 lemma sumI [intro]:

     "u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> x = u + v \<Longrightarrow> x \<in> U + V"

   unfolding sum_def by blast

 lemma sumI' [intro]:

     "u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> u + v \<in> U + V"

   unfolding sum_def by blast

 text {* @{text U} is a subspace of @{text "U + V"}. *}

 lemma subspace_sum1 [iff]:

   assumes "vectorspace U" "vectorspace V"

   shows "U \<unlhd> U + V"

 proof -

   interpret vectorspace U by fact

   interpret vectorspace V by fact

   show ?thesis

   proof

     show "U \<noteq> {}" ..

     show "U \<subseteq> U + V"

     proof

       fix x assume x: "x \<in> U"

       moreover have "0 \<in> V" ..

       ultimately have "x + 0 \<in> U + V" ..

       with x show "x \<in> U + V" by simp

     qed

     fix x y assume x: "x \<in> U" and "y \<in> U"

     then show "x + y \<in> U" by simp

     from x show "\<And>a. a \<cdot> x \<in> U" by simp

   qed

 qed

 text {* The sum of two subspaces is again a subspace. *}

 lemma sum_subspace [intro?]:

   assumes "subspace U E" "vectorspace E" "subspace V E"

   shows "U + V \<unlhd> E"

 proof -

   interpret subspace U E by fact

   interpret vectorspace E by fact

   interpret subspace V E by fact

   show ?thesis

   proof

     have "0 \<in> U + V"

     proof

       show "0 \<in> U" using `vectorspace E` ..

       show "0 \<in> V" using `vectorspace E` ..

       show "(0::'a) = 0 + 0" by simp

     qed

     then show "U + V \<noteq> {}" by blast

     show "U + V \<subseteq> E"

     proof

       fix x assume "x \<in> U + V"

       then obtain u v where "x = u + v" and

 	"u \<in> U" and "v \<in> V" ..

       then show "x \<in> E" by simp

     qed

     fix x y assume x: "x \<in> U + V" and y: "y \<in> U + V"

     show "x + y \<in> U + V"

     proof -

       from x obtain ux vx where "x = ux + vx" and "ux \<in> U" and "vx \<in> V" ..

       moreover

       from y obtain uy vy where "y = uy + vy" and "uy \<in> U" and "vy \<in> V" ..

       ultimately

       have "ux + uy \<in> U"

 	and "vx + vy \<in> V"

 	and "x + y = (ux + uy) + (vx + vy)"

 	using x y by (simp_all add: add_ac)

       then show ?thesis ..

     qed

     fix a show "a \<cdot> x \<in> U + V"

     proof -

       from x obtain u v where "x = u + v" and "u \<in> U" and "v \<in> V" ..

       then have "a \<cdot> u \<in> U" and "a \<cdot> v \<in> V"

 	and "a \<cdot> x = (a \<cdot> u) + (a \<cdot> v)" by (simp_all add: distrib)

       then show ?thesis ..

     qed

   qed

 qed

 text{* The sum of two subspaces is a vectorspace. *}

 lemma sum_vs [intro?]:

     "U \<unlhd> E \<Longrightarrow> V \<unlhd> E \<Longrightarrow> vectorspace E \<Longrightarrow> vectorspace (U + V)"

   by (rule subspace.vectorspace) (rule sum_subspace)

 subsection {* Direct sums *}

 text {*

   The sum of @{text U} and @{text V} is called \emph{direct}, iff the

   zero element is the only common element of @{text U} and @{text

   V}. For every element @{text x} of the direct sum of @{text U} and

   @{text V} the decomposition in @{text "x = u + v"} with

   @{text "u \<in> U"} and @{text "v \<in> V"} is unique.

 *}

 lemma decomp:

   assumes "vectorspace E" "subspace U E" "subspace V E"

   assumes direct: "U \<inter> V = {0}"

     and u1: "u1 \<in> U" and u2: "u2 \<in> U"

     and v1: "v1 \<in> V" and v2: "v2 \<in> V"

     and sum: "u1 + v1 = u2 + v2"

   shows "u1 = u2 \<and> v1 = v2"

 proof -

   interpret vectorspace E by fact

   interpret subspace U E by fact

   interpret subspace V E by fact

   show ?thesis

   proof

     have U: "vectorspace U"  (* FIXME: use interpret *)

       using `subspace U E` `vectorspace E` by (rule subspace.vectorspace)

     have V: "vectorspace V"

       using `subspace V E` `vectorspace E` by (rule subspace.vectorspace)

     from u1 u2 v1 v2 and sum have eq: "u1 - u2 = v2 - v1"

       by (simp add: add_diff_swap)

     from u1 u2 have u: "u1 - u2 \<in> U"

       by (rule vectorspace.diff_closed [OF U])

     with eq have v': "v2 - v1 \<in> U" by (simp only:)

     from v2 v1 have v: "v2 - v1 \<in> V"

       by (rule vectorspace.diff_closed [OF V])

     with eq have u': " u1 - u2 \<in> V" by (simp only:)

     show "u1 = u2"

     proof (rule add_minus_eq)

       from u1 show "u1 \<in> E" ..

       from u2 show "u2 \<in> E" ..

       from u u' and direct show "u1 - u2 = 0" by blast

     qed

     show "v1 = v2"

     proof (rule add_minus_eq [symmetric])

       from v1 show "v1 \<in> E" ..

       from v2 show "v2 \<in> E" ..

       from v v' and direct show "v2 - v1 = 0" by blast

     qed

   qed

 qed

 text {*

   An application of the previous lemma will be used in the proof of

   the Hahn-Banach Theorem (see page \pageref{decomp-H-use}): for any

   element @{text "y + a \<cdot> x\<^sub>0"} of the direct sum of a

   vectorspace @{text H} and the linear closure of @{text "x\<^sub>0"}

   the components @{text "y \<in> H"} and @{text a} are uniquely

   determined.

 *}

 lemma decomp_H':

   assumes "vectorspace E" "subspace H E"

   assumes y1: "y1 \<in> H" and y2: "y2 \<in> H"

     and x': "x' \<notin> H"  "x' \<in> E"  "x' \<noteq> 0"

     and eq: "y1 + a1 \<cdot> x' = y2 + a2 \<cdot> x'"

   shows "y1 = y2 \<and> a1 = a2"

 proof -

   interpret vectorspace E by fact

   interpret subspace H E by fact

   show ?thesis

   proof

     have c: "y1 = y2 \<and> a1 \<cdot> x' = a2 \<cdot> x'"

     proof (rule decomp)

       show "a1 \<cdot> x' \<in> lin x'" ..

       show "a2 \<cdot> x' \<in> lin x'" ..

       show "H \<inter> lin x' = {0}"

       proof

 	show "H \<inter> lin x' \<subseteq> {0}"

 	proof

           fix x assume x: "x \<in> H \<inter> lin x'"

           then obtain a where xx': "x = a \<cdot> x'"

             by blast

           have "x = 0"

           proof cases

             assume "a = 0"

             with xx' and x' show ?thesis by simp

           next

             assume a: "a \<noteq> 0"

             from x have "x \<in> H" ..

             with xx' have "inverse a \<cdot> a \<cdot> x' \<in> H" by simp

             with a and x' have "x' \<in> H" by (simp add: mult_assoc2)

             with `x' \<notin> H` show ?thesis by contradiction

           qed

           then show "x \<in> {0}" ..

 	qed

 	show "{0} \<subseteq> H \<inter> lin x'"

 	proof -

           have "0 \<in> H" using `vectorspace E` ..

           moreover have "0 \<in> lin x'" using `x' \<in> E` ..

           ultimately show ?thesis by blast

 	qed

       qed

       show "lin x' \<unlhd> E" using `x' \<in> E` ..

     qed (rule `vectorspace E`, rule `subspace H E`, rule y1, rule y2, rule eq)

     then show "y1 = y2" ..

     from c have "a1 \<cdot> x' = a2 \<cdot> x'" ..

     with x' show "a1 = a2" by (simp add: mult_right_cancel)

   qed

 qed

 text {*

   Since for any element @{text "y + a \<cdot> x'"} of the direct sum of a

   vectorspace @{text H} and the linear closure of @{text x'} the

   components @{text "y \<in> H"} and @{text a} are unique, it follows from

   @{text "y \<in> H"} that @{text "a = 0"}.

 *}

 lemma decomp_H'_H:

   assumes "vectorspace E" "subspace H E"

   assumes t: "t \<in> H"

     and x': "x' \<notin> H"  "x' \<in> E"  "x' \<noteq> 0"

   shows "(SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H) = (t, 0)"

 proof -

   interpret vectorspace E by fact

   interpret subspace H E by fact

   show ?thesis

   proof (rule, simp_all only: split_paired_all split_conv)

     from t x' show "t = t + 0 \<cdot> x' \<and> t \<in> H" by simp

     fix y and a assume ya: "t = y + a \<cdot> x' \<and> y \<in> H"

     have "y = t \<and> a = 0"

     proof (rule decomp_H')

       from ya x' show "y + a \<cdot> x' = t + 0 \<cdot> x'" by simp

       from ya show "y \<in> H" ..

     qed (rule `vectorspace E`, rule `subspace H E`, rule t, (rule x')+)

     with t x' show "(y, a) = (y + a \<cdot> x', 0)" by simp

   qed

 qed

 text {*

   The components @{text "y \<in> H"} and @{text a} in @{text "y + a \<cdot> x'"}

   are unique, so the function @{text h'} defined by

   @{text "h' (y + a \<cdot> x') = h y + a \<cdot> \<xi>"} is definite.

 *}

 lemma h'_definite:

   fixes H

   assumes h'_def:

     "h' \<equiv> (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a \<cdot> x' \<and> y \<in> H)

                 in (h y) + a * xi)"

     and x: "x = y + a \<cdot> x'"

   assumes "vectorspace E" "subspace H E"

   assumes y: "y \<in> H"

     and x': "x' \<notin> H"  "x' \<in> E"  "x' \<noteq> 0"

   shows "h' x = h y + a * xi"

 proof -

   interpret vectorspace E by fact

   interpret subspace H E by fact

   from x y x' have "x \<in> H + lin x'" by auto

   have "\<exists>!p. (\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> H) p" (is "\<exists>!p. ?P p")

   proof (rule ex_ex1I)

     from x y show "\<exists>p. ?P p" by blast

     fix p q assume p: "?P p" and q: "?P q"

     show "p = q"

     proof -

       from p have xp: "x = fst p + snd p \<cdot> x' \<and> fst p \<in> H"

         by (cases p) simp

       from q have xq: "x = fst q + snd q \<cdot> x' \<and> fst q \<in> H"

         by (cases q) simp

       have "fst p = fst q \<and> snd p = snd q"

       proof (rule decomp_H')

         from xp show "fst p \<in> H" ..

         from xq show "fst q \<in> H" ..

         from xp and xq show "fst p + snd p \<cdot> x' = fst q + snd q \<cdot> x'"

           by simp

       qed (rule `vectorspace E`, rule `subspace H E`, (rule x')+)

       then show ?thesis by (cases p, cases q) simp

     qed

   qed

   then have eq: "(SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H) = (y, a)"

     by (rule some1_equality) (simp add: x y)

   with h'_def show "h' x = h y + a * xi" by (simp add: Let_def)

 qed

 end